Use the theorems that you that you have learned about inscribed angles and intercepted arcs to explain why inscribed quadrilaterals have opposite angles that are supplementary. Answer in 5-6 lines.
Since you have learned about inscribed angles and intercepted arcs, it is safe to assume that you are familiar with what they are.
Following is a summary of hints to help complete the proof.
fact #1: the sum of the intercepted arcs of the sides of a cyclic quadrilateral (a quad inscribed in a circle) adds up to 360°. Draw a sketch to convince yourself of this fact.
fact #2: The inscribed angle subtended by each side is exactly half the intercepted arc subtended by the same side.
The final step:
Draw a quad inscribed in a circle, then draw exactly one diagonal. Now identify all the inscribed angles, and determine what the sum of the four angles are, using facts 1 and 2.
If you need to review what inscribed angles and intercepted arcs are, try
http://www.onlinemathlearning.com/arc-angles.html
or
http://www.mathopenref.com/arcintercepted.html
To understand why inscribed quadrilaterals have opposite angles that are supplementary, we need to use the theorems related to inscribed angles and intercepted arcs. The first theorem states that an inscribed angle is half the measure of the intercepted arc. Therefore, each angle of the quadrilateral corresponds to an intercepted arc.
Now, consider the opposite angles of the inscribed quadrilateral. Let's call them Angle A and Angle C and the corresponding intercepted arcs Arc AB and Arc CD, respectively. By the theorem, Angle A is half of Arc AB, and Angle C is half of Arc CD.
Since the sum of angles in a quadrilateral is 360 degrees, we have Angle A + Angle C = Arc AB/2 + Arc CD/2. Simplifying this expression, we get Angle A + Angle C = (Arc AB + Arc CD)/2. But by the property of intercepted arcs, Arc AB + Arc CD = 360 degrees.
Therefore, Angle A + Angle C = 360/2 = 180 degrees. Hence, the opposite angles of an inscribed quadrilateral are supplementary.